In mathematics, Busemann's theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

and geometric tomography. It was first proved by

Herbert Busemann Herbert Busemann (12 May 1905 – 3 February 1994) was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemann's theorem in Euclidean geometry and geometric tomography. He was a member ...

in 1949 and was motivated by his theory of area in Finsler spaces.

Statement of the theorem

Let ''K'' be a

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...

in ''n''-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

R^''n'' containing the

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

in its interior. Let ''S'' be an (''n'' − 2)-dimensional linear subspace of R^''n''. For each

unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...

''θ'' in ''S''^⊥, the

orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

of ''S'', let ''S''_''θ'' denote the (''n'' − 1)-dimensional

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...

containing ''θ'' and ''S''. Define ''r''(''θ'') to be the (''n'' − 1)-dimensional volume of ''K'' ∩ ''S''_''θ''. Let ''C'' be the curve in ''S''^⊥. Then ''C'' forms the boundary of a convex body in ''S''^⊥.

References

* * {{cite journal , last=Gardner , first=Richard J. , title=The Brunn-Minkowski inequality , journal=Bull. Amer. Math. Soc. (N.S.) , volume=39 , issue=3 , year=2002 , pages=355–405 (electronic) , doi=10.1090/S0273-0979-02-00941-2 , citeseerx=10.1.1.106.7344 Euclidean geometry Geometric inequalities Theorems in convex geometry

Statement of the theorem

See also

References